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Section 6.6 Chapter Review

We now have covered all of the equilibrium types in the table below. You should be able to compare and contrast the various types of equilibrium, draw the free-body diagrams associated with each, and know which equations allow you to solve for the unknowns of each case.

This table compares the various types of equilibrium topics, plus solving techniques and number of equations available for each.

Table 6.6.1. Equilibrium Methods

System Type

Definition

Technique to Solve

Equations and Unknowns

Particle (Ch. 3)

A single particle where all forces (external, body, and reactions) are all collinear and in equilibrium

Collinear forces allow us to isolate particle as free-body diagram (cutting axial forces in 2-force supports) and \(\Sigma \vec{F} = 0 \)

(Note: \(\Sigma \vec{M} = 0\) not helpful as forces do not create a moment about particle)

2D →

\begin{align*} 2\; \Sigma F\amp =0 \text{ equations} \\ \amp = 2 \text{ unknowns} \end{align*}

3D →

\begin{align*} 3\; \Sigma F\amp =0 \text{ equations} \\ \amp = 3 \text{ unknowns} \end{align*}

Rigid Body (Ch. 5)

A multi-force member where external loading and reactions are along various lines of action and are in equilibrium

A variety of forces and couples (from loading and reactions) require a free-body diagram of a rigid body and apply \(\Sigma \vec{F} = 0\) and \(\Sigma \vec{M} = 0\) equations.

2D →

\begin{align*} 2\; \Sigma F\amp =0 \\ + 1\; \Sigma M\amp =0 \text{ equations} \\ \amp = 3 \text{ unknowns} \end{align*}

3D →

\begin{align*} 3\;\Sigma F\amp =0 \\ + 3\; \Sigma M\amp =0 \text{ equations} \\ \amp = 6 \text{ unknowns} \end{align*}

Truss (Ch. 6)

A multiple rigid body system structure that consists entirely of two-force members, and only carries forces at joints between members

Two-force members and loading at joints allow free-body diagram of joints to expose axial loads in members

2D →

\begin{align*} 2\; \Sigma F\amp =0 \text{ equations per joint} \\ \amp = (\text{reactions}) + (\text{members}) \end{align*}

3D →

\begin{align*} 3\; \Sigma F\amp =0 \text{ equations per joint}) \\ \amp = (\text{reactions}) + (\text{members}) \end{align*}

Frame (Ch. 6)

A multiple rigid body system that is designed not to move and contains at least one member that is not a two-force member.

The mix of two-force and multiforce bodies require free-body diagrams of rigid bodies where interactions between bodies are equal and opposite (as per Newton’s 3rd Law). Can apply \(\Sigma F=0\) and \(\Sigma M=0 \)equations for each body to solve for total unknowns.

2D →

\begin{align*} 2\;\Sigma F\amp =0 \\ + 1\; \Sigma M\amp =0 \text{ equations per multiforce body} \\ + 1\; \Sigma F\amp =0 \text{ equation per two-force body} \end{align*}

3D →

\begin{align*} (3\; \Sigma F\amp =0 \\ + 3\; \Sigma M\amp =0 \text{ equations per multiforce body} \\ + 1\; \Sigma F\amp =0 \text{ equation per two-force body} \end{align*}

Machine (Ch. 6)

A multiple rigid body system where the parts can move relative to one another and that contains at least one member that is not a two-force member.

** Other combinations of equilibrium equations are possible see Ch. 5 Rigid Body Equilibrium Section XX for details